A non-circular Eshelby inclusion near a non-parabolic inhomogeneity admitting internal uniform stresses

With the aid of conformal mapping and analytic continuation, we prove that within the framework of anti-plane elasticity, a non-parabolic open elastic inhomogeneity can still admit an internal uniform stress field despite the presence of a nearby non-circular Eshelby inclusion undergoing uniform anti-plane eigenstrains when the surrounding elastic matrix is subjected to uniform remote stresses. The non-circular inclusion can take the form of a Booth’s lemniscate inclusion, a generalized Booth’s lemniscate inclusion or a cardioid inclusion. Our analysis indicates that the uniform stress field within the non-parabolic inhomogeneity is independent of the specific open shape of the inhomogeneity and is also unaffected by the existence of the nearby non-circular inclusion. On the other hand, the non-parabolic shape of the inhomogeneity is caused solely by the presence of the non-circular inclusion.

Sensitivity of the Second Order Homogenized Elasticity Tensor to Topological Microstructural Changes

The multiscale elasticity model of solids with singular geometrical perturbations of microstructure is considered for the purposes, e.g., of optimum design. The homogenized linear elasticity tensors of first and second orders are considered in the framework of periodic Sobolev spaces. In particular, the sensitivity analysis of second order homogenized elasticity tensor to topological microstructural changes is performed. The derivation of the proposed sensitivities relies on the concept of topological derivative applied within a multiscale constitutive model. The microstructure is topologically perturbed by the nucleation of a small circular inclusion that allows for deriving the sensitivity in its closed form with the help of appropriate adjoint states. The resulting topological derivative is given by a sixth order tensor field over the microstructural domain, which measures how the second order homogenized elasticity tensor changes when a small circular inclusion is introduced at the microscopic level. As a result, the topological derivatives of functionals for multiscale models can be obtained and used in numerical methods of shape and topology optimization of microstructures, including synthesis and optimal design of metamaterials by taking into account the second order mechanical effects. The analysis is performed in two spatial dimensions however the results are valid in three spatial dimensions as well.

GPU-based pseudo-transient finite difference solution for power-law viscous flow in cartesian, polar and spherical coordinates

<p>Power-law viscous flow describes well the first-order features of long-term lithosphere deformation. Due to the ellipticity of the Earth, the lithosphere is mechanically analogous to a shell, characterized by a double curvature. The mechanical characteristics of a shell are fundamentally different to the characteristics of plates, having no curvature in their undeformed state. The systematic quantification of the magnitude and the spatiotemporal distribution of strain, strain-rate and stress inside a deforming lithospheric shell is thus of major importance: stress is for example a key physical quantity that controls geodynamic processes such as metamorphic reactions, decompression melting, lithospheric flexure, subduction initiation or earthquakes.</p><p>Stress calculations in a geometrically and mechanically heterogeneous 3-D lithospheric shell require high-resolution and high-performance computing. The pseudo-transient finite difference (PTFD) method recently enabled efficient simulations of high-resolution 3-D deformation processes, implementing an iterative implicit solution strategy of the governing equations for power-law viscous flow. Main challenges for the PTFD method is to guarantee convergence, minimize the required iteration count and speed-up the iterations.</p><p>Here, we present PTFD simulations for simple mechanically heterogeneous (weak circular inclusion) incompressible 2-D power-law viscous flow in cartesian and cylindrical coordinates. The flow laws employ a pseudo-viscoelastic behavior to optimize the iterative solution by exploiting the fundamental characteristics of viscoelastic wave propagation.</p><p>The developed PTFD algorithm executes in parallel on CPUs and GPUs. The development was done in Matlab (mathworks.com), then translated into the Julia language (julialang.org), and finally made compatible for parallel GPU architectures using the ParallelStencil.jl package (https://github.com/omlins/ParallelStencil.jl). We may unveil preliminary results for 3-D spherical configurations including gravity-controlled lithospheric stress distributions around continental plateaus.</p>

An Interface Anticrack in a Periodic Two–Layer Piezoelectric Space under Vertically Uniform Heat Flow

This paper aims to investigate 3D static thermoelectroelastic problem of a uniform heat flow in a bi-material periodically layered space disturbed by a thermally and electrically-insulated rigid sheet-like inclusion (so-called anticrack) situated at one of the interfaces. An approximate analysis of the considered laminated composite is given in the framework of the homogenized model with microlocal parameters. Accurate results are obtained by constructing suitable potential solutions and reducing to the corresponding homogeneous thermoelectromechanical (or thermomechanical) anticrack problems. The governing boundary integral equation for a planar interface anticrack of arbitrary shape is derived in terms of a normal stress discontinuity. As an illustration, a complete solution for a rigid circular inclusion is obtained in terms of elementary functions and interpreted from the failure perspective. Unlike existing solutions for defects at the interface of materials, the solution obtained displays no oscillatory behavior.

A Circular Inclusion and Two Radial Coaxial Cracks with Contacting Faces in a Piecewise Homogeneous Isotropic Plate Under Bending

The bending problem of an infinite, piecewise homogeneous, isotropic plate with circular interfacial zone and two coaxial radial cracks is solved on the assumption of crack closure along a line on the plate surface. Using the theory of functions of a complex variable, complex potentials and a superposition of plane problem of the elasticity theory and plate bending problem, the solution is obtained in the form of a system of singular integral equations, which is numerically solved after reducing to a system of linear algebraic equations by the mechanical quadrature method. Numerical results are presented for the forces and moments intensity factors, contact forces between crack faces and critical load for various geometrical and mechanical task parameters.